One-parameter Family of Radon-Nikodym Theorems for States of a von Neumann Algebra

It is shown that any normal state (p of a von Neumann algebra 972 with a cyclic and separating vector W satisfying (p^.la)¥ for some />0 has a representative vector 0a in Fp for each ae [0,1/2] and 0a = Qa¥ for a Qae9# satisfying ||QJ|^/ when ae[0,1/4], §1. Male Theorem Let W be a von Neumann algebra on a Hilbert space § with a unit cyclic and separating vector W. Let Aw be the modular operator for 991, of $Jl such that (p^la>T for some />0, there exists a vector ^aeF^/ for every ae[0, 1/2] such that Combined with Theorem 3(8) of [2], Theorem 1 implies the following: Theorem 2. For any normal state cp of %R such that (prgko^, there exists a gae9Jl for a 6 [0,1/4] such that Q}Qatp=(p, ||QJ^1 . Operators ga, such that Qa*F e F^, are characterized in Theorem 3(7) of [2] by the property that of (ga) has an analytic continuation Received July 13, 1973.